The main focus of our work is on the coalgebras of the functor \(\mathcal {L}\)  on
\(\mathsf {SMod}\) , the category of semimodules over the semiring \(K\) . We start by
defining the functor \(\mathcal {L}\)  and proceed to define the notions of \(K\) -linear
automata as coalgebras for the functor \(\mathcal {L}\) . We define their behaviour in terms
of weighted languages and present the final \(\mathcal {L}\) -coalgebra
(Section REF ). This part is more or less similar to the treatment
in [1]} as it does not rely on any vector space property
not enjoyed by a semimodule. The point of departure in our work from that of
[1]} are the definitions of \(K\) -linear relations and
\(K\) -linear bisimulations (Section REF ) based on the kernel (in the
universal algebraic sense) of \(K\) -linear maps as mentioned above. We prove the
correspondence between \(K\) -linear bisimulations and kernels of
\(\mathcal {L}\) -homomorphisms and establish the coincidence of \(\approx _\mathcal {L}\)  and \(\sim _l\) 
(Section REF ). Although these results mirror those of
[1]}, the proofs, other than the one for the coincidence
of \(\approx _\mathcal {L}\)  and \(\sim _l\) , are new and more general – they truly extend the
results from the vector space setting to the semimodule setting in a non-trivial
way. This is where the main contribution of this paper lies.
